Raw score to measure conversion table

The score-to-logit conversion for complete responders: every possible raw score with its location, standard error, and the frequency and cumulative percentage of complete responders at that score (the complete-data estimates table of Andrich and Marais 2019, ch. 10).

Usage

score_table(
  fit,
  method = c("wle", "mle"),
  extremes = c("model", "extrapolated")
)

Arguments

fit

A fitted object from rasch.

method

"wle" (Warm, default) or "mle".

extremes

Treatment of the extreme scores. "model" keeps the estimator's own values; these are NA for MLE. "extrapolated" applies the geometric extrapolation.

Value

A data frame with score, theta, se, freq, cum_pct (omitted when no complete responders exist), and extrapolated; NULL when the fitted items do not share one discrimination or an item is represented by several MFRM or EFRM response cells.

Details

Two estimators are available. "wle" (the default) is Warm's weighted likelihood estimate, finite at the extreme scores. "mle" is the plain maximum likelihood estimate, infinite at the extremes. extremes = "extrapolated" replaces the extreme-score entries by the geometric extrapolation described in Andrich and Marais (2019, ch. 10): successive score-to-score differences grow towards the extremes, so the last difference is continued geometrically – the extrapolated top difference \(d\) solves \(b = \sqrt{a d}\) where \(a, b\) are the two preceding differences (equivalently \(d = b^2/a\)), and symmetrically at zero. The standard error at an extrapolated location is \(1/\sqrt{I(\theta)}\) evaluated there. With method = "wle" the extrapolation replaces the finite Warm estimates at the extremes, giving the extrapolated form of the conversion table from a WLE analysis.

References

Andrich, D. and Marais, I. (2019). A Course in Rasch Measurement Theory: Measuring in the Educational, Social and Health Sciences. Springer.

Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54(3), 427–450.

Examples

set.seed(1)
d <- seq(-1.5, 1.5, length.out = 6)
X <- matrix(rbinom(300 * 6, 1, plogis(outer(rnorm(300), d, "-"))), 300, 6)
colnames(X) <- paste0("I", 1:6)
score_table(rasch(X), method = "mle", extremes = "extrapolated")
#>  score  theta    se extrapolated freq cum_pct
#>      0 -3.278 1.825            *   13   4.333
#>      1 -1.954 1.160                35  16.000
#>      2 -0.881 0.958                63  37.000
#>      3 -0.011 0.922                73  61.333
#>      4  0.868 0.967                66  83.333
#>      5  1.963 1.170                41  97.000
#>      6  3.326 1.848            *    9 100.000